About the Lesson
- This lesson involves the product of complex numbers, powers of
, i and complex conjugates.
- As a result students will:
- Compute the product of two complex numbers and write a
general symbolic result for the product.
- Discover the pattern in powers of i and simplify i raised to
any power.
- Compute the product of complex conjugates and relate this to
the absolute value of the original complex number.
- Visualize the product of complex numbers written in polar
form and write the product in polar form.

TI-Nspire™ Navigator™ System
- Transfer a File.
- Use Screen Capture to examine patterns that emerge.
- Use Live Presenter to demonstrate.
- Use Quick Poll to assess students’ understanding.

TI-Nspire™ Technology Skills:
- Download a TI-Nspire
document
- Open a document
- Move between pages
- Grab and drag a point

Tech Tips:
- Make sure the font size on
your TI-Nspire handhelds is
set to Medium.
- You can hide the function
entry line by pressing /
G.

Tech Tip: If students experience difficulty dragging a point, check to make
sure that they have moved the cursor until it becomes a hand (÷) getting
ready to grab the point. Also, be sure that the word point appears, not the
word text. Then press / x to grab the point and close the hand ({).
There are several interactive Math Boxes in this activity. Remember that if
you change a definition in one Math Box, the handheld will automatically
update all other interactive Math Boxes related to this change.

Move to page 1.2.

1. This Notes Page contains three interactive Math Boxes for the
complex numbers z and w and the product · . p z w =
a. Redefine z and/or w as necessary to complete the
following tables. Note: to redefine z or w, edit the Math
Box following the assignment characters, := .

Answer: Note: To access i on the handheld, press ¹ to obtain a list of special mathematical
symbols. Use the arrow keys to highlight i and press ·.

TI-Nspire Navigator Opportunity: Screen Capture
See Note 1 at the end of this lesson.

Move to page 1.3.

2. This Notes Page contains two interactive Math Boxes to
produce a sequence of the powers of 1. i = ÷ That is, it
constructs a sequence of the form
2 3
, , , , .
end
i i i i . The variable
end is the largest value the sequence variable will assume, in
this case the last power of . i .
a. Change the value of the variable end as necessary to
complete the following tables. Note: the variable end is
defined in a Math Box. To redefine the value of end, edit the
Math Box following the assignment characters, := .

TI-Nspire Navigator Opportunity: Quick Poll
See Note 2 at the end of this lesson.

Move to page 1.4.

3. This Notes Page contains three interactive Math Boxes for the
complex number , z its complex conjugate cz (denoted z ), and
the product · . z cz
a. Change the value of z as necessary to complete the
following table. To change the value of z , edit the Math
Box following the assignment characters, : . =

c. Recall that a complex number can be represented by a point in the complex plane.
For z a bi = + find · r z z = and interpret this value.

Answer:
2 2
· r z z a b = = +
This value is the absolute value, or magnitude, of the complex number . z The product of complex
conjugates z and z is the square of the distance from the origin to the point representing z in
the complex plane. Note: the product of a complex number and its conjugate is a real number.

Move to page 2.1.

For any complex number z a bi = + , the absolute value, or
magnitude, is
2 2
| | . r z a b = = + The absolute value is the
distance from the origin to the point representing z in the complex
plane. The argument of the complex number , z arg( ), z is the
angle u (in radians) formed between the positive real axis and the
position vector representing . z The angle is positive if measured
counterclockwise from the positive real axis.

4. On Page 2.1, the complex number z is represented by a point and a position vector. The value of
, z the absolute value, and the argument are given on this page. Drag and position z as necessary
to answer the following questions.
Describe the location of the point representing z in the complex plane if:
a. 2 r = and
4
t
u =

Sample Answers: The point representing z lies in the first quadrant, 2 units from the origin, and
on the ray from the origin that makes an angle
4
t
with the positive real axis.
Note: It might be difficult for students to move the point z in order to obtain the exact values of r
and . u Students will also have to convert the symbolic value of u to a decimal approximation.

Sample Answer: The point representing z lies in the second quadrant, 4 units from the
origin, and on the ray from the origin that makes an angle
5
6
t
with the positive real axis.

c. 1 r = and
4
3
t
u = ÷

Sample Answer: The point representing z lies in the third quadrant, 1 unit from the origin,
and on the ray from the origin that makes an angle
4
3
t
÷ with the positive real axis. Note: this
point lies on the unit circle.

d. 3 r = and
13
4
t
u =

Sample Answer: The point representing z lies in the third quadrant, 3 units from the origin,
and on the ray from the origin that makes an angle
13
4
t
or
5
4
t
with the positive real axis.

e. 3 r = and u t =

Sample Answer: The point representing z lies on the real axis and 3 units to the left of the
origin (on the ray from the origin that makes an angle t with the positive real axis).

f. 2 r =

Sample Answer: This equation describes infinitely many points in the complex plane that lie
on the circle of radius 2 which is centered at the origin.

g.
3
2
t
u =

Sample Answer: This equation describes infinitely many points in the complex plane that lie
on the ray from the origin that makes an angle
3
2
t
with the positive real axis.

Any complex number z a bi = + with | | r z = and arg( ) z u = can
be written in polar form as (cos sin ). z r i u u = + Page 3.2
illustrates the product of two complex numbers in polar form.
5. The complex numbers , z , w and p are represented by triangles.
When you drag either the point z or the point w, the product is
automatically computed, and the triangle representing p is
updated. Note that a copy of the triangle representing z is
rotated so that the vertex lies along the hypotenuse of the triangle
that represents . w Move z and w around the first quadrant, and
observe the absolute value and argument for the three complex
numbers.

a. Write an equation which seems to define
3
u in terms of
1
u and
2
. u

On Page 4.1, click on the arrows to step through the process of
multiplication. This figure might provide further insight about the
relationship among the absolute values and arguments. Right-click
in the slider box, and select option 3 to animate this figure.

To obtain the product in polar form, multiply the absolute values, and add the arguments.

Extensions

1. For a complex number z in polar form, ask students to write a formula for
n
z and
n
z in polar form.
2. For two complex numbers z and w in polar form, ask students to write a formula for
z
w
in polar
form.
3. Ask students to describe the point(s) representing z in the complex plane if 0 r < and/or 0. u <

Wrap Up
Upon completion of the lesson, the teacher should ensure that students are able to understand:
- How to multiply two complex numbers in a bi + form.
- How to multiply two complex numbers in polar form.
- The cyclic nature of powers of . i
- The relationship between the product of complex conjugates and the absolute value of the
original complex number.